Bifurcation Analysis And Applications Of Systems With State–dependent Delays

Over the past several years,differential equations with state-dependent de-lays(SD-DDEs)have been applied in many fields such as electrodynamics,popu-lation growth,economics,engineering,neural network,network congestion control and so on.However,there are a lot of challenges in the study of SD-DDEs due to its weak smoothness of the solution space which is very different from differential equation with constant time delays.It also makes the study of fundamental the-ory of SD-DDEs more complicated.And there are also few analytical method for the dynamics and bifurcations in SD-DDEs.Thus,it is very necessary to explore some new and valid techniques to study the dynamical phenomena in SD-DDEs.This thesis focuses on exploring some effective mathematical methods to study the complicated dynamics of SD-DDEs.We mainly try to extend the method of multiple scale(MMS)and harmonic balance method with alternat-ing frequency/time domain technique(HB-AFT)to differential equations with state-dependent time delays and study the complicated dynamics and the analyt-ical expression of period solution.This is a very useful work and has important guiding significance to practical application.The main works of the present thesis are listed in the following.(1)The development and applications of SD-DDEs since 2006 are reviewed in detail.Firstly,we summarize the basic theory of SD-DDEs developed in recent years,such as the framework of initial value problem,linearization and stability,invariant manifold of solution,periodic solution and Hopf bifurcation theory and so on.Then,the applications of SD-DDEs on mathematical modeling,stability and bifurcation are introduced in different field.Finally,we present some prospects about the study of SD-DDEs.(2)MMS is extended to the study of SD-DDEs.Hopf bifurcation and double Hopf bifurcation of a simple linear paradigm of SD-DDEs are studied in detail based on formal linearization,stability theory,MMS and normal form.The comparison between numerical and analytical results indicate that MMS is valid and accurate.(3)MMS is applied to explore the complicated oscillation mechanism in the gene expression process.In order to explain the oscillatory phenomena in the gene expression process,Hopf bifurcation and double Hopf bifurcation of gene expression models with state-dependent delay are investigated based on MMS and normal form analysis.Besides,rich and complicated oscillations,such as periodic solution,quasi-periodic solution,period-2 solution are also explored in this thesis.And it is very useful to understand the complicated oscillation mechanism in the gene expression process.(4)HB-AFT is extended to the study of periodic solution in differential equation with state-dependent delay and non-smooth function.The periodic solution-s of both the source dynamics and queue dynamics in network congestion control system are considered in detail employing the HB-AFT for the first time.And complicated dynamics such as bistable periodic solution,period-m solution,chaos and multi-stability,are displayed by numerical bifurcation analysis.It can be seen that there are a lot of rich and complicated dynamics in the network congestion control system.Thus,the control and parameters tuning based on the above results could be used to avoid the network conges-tion and optimize its performance.And it also gives an important guidance to practical application in network congestion control.The main highlights of this thesis are as follows:1.MMS is successfully extended to the study of differential equations with state-dependent delays.2.HB-AFT is extended to differential equations with state-dependent delays and non-smooth function and the analytical approximate expressions of periodic solutions of them are obtained for the first time.3.Complicated dynamics such as period-m solution,torus,phase-locked so-lution,chaos,multi-stability and so on,are discovered in SD-DDEs.